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A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For a geometric series to be effectively analyzed, it usually needs a recognizable pattern such as:

• The first term, denoted as \(a\)
• A common ratio, denoted as \(r\)

To express the repeating decimal 0.090909... as a geometric series, we need to look at it as an infinite series where each term is derived by multiplying the previous term by a certain factor (the common ratio). In this case, each subsequent term is the previous term multiplied by \(10^{-2}\). By identifying the repeating pattern — 09 in the tenths and hundredths place — we can express the decimal as:\[0.090909\ldots = 9(10^{-2} + 10^{-4} + 10^{-6} + \ldots)\]This shows clearly how the repeated sequence of the decimal fits into the framework of a geometric series.

Calculating the sum of an infinite geometric series like 0.090909... requires understanding a special formula. For an infinite geometric series to have a sum, the absolute value of the common ratio \(r\) must be less than 1.The formula to find the sum \(S\) of such a series is:\[S = \frac{a}{1 - r}\]where \(a\) is the first term and \(r\) is the common ratio. In the given example, the first term \(a = 9 \times 10^{-2}\) and the common ratio \(r = 10^{-2}\). Plugging these into the formula, we calculate:\[S = \frac{10^{-2}}{1 - 10^{-2}}\]This formula helps sum up all the infinitely many terms by accounting for their rapidly decreasing sizes. Hence, multiplying the result by the coefficient 9 accounts for the repeating portion throughout the decimal sequence, ultimately allowing us to express the infinite decimal as a finite fraction.

Converting a repeating decimal to a fraction involves using the concept and result from the geometric series. Once the infinite series is expressed and summed as described earlier, the repeating decimal can be rewritten as a simple fraction.First, multiply the sum of the infinite series by the coefficient, which represents how many times the fractional unit (formed from the series) repeats. For 0.090909..., this coefficient is 9:\[0.090909\ldots = 9 \cdot \frac{10^{-2}}{1 - 10^{-2}}\]Then, simplify this complex expression by rationalizing it. By multiplying both the numerator and the denominator with the same power of ten that clears the denominators, the fraction simplifies to:\[0.090909\ldots = \frac{9}{99}\]The final step is simplifying \(\frac{9}{99}\). Both the numerator and denominator can be divided by their greatest common divisor, which is 9, yielding:\[\frac{9}{99} = \frac{1}{11}\]Thus, the repeating decimal \(0.\overline{09}\) expressed as a proper fraction is \(\frac{1}{11}\), completing our conversion efficiently.